Understanding CNN in a few sentences
To address your description point by point,


There are many types of CNN architectures. You appear to be describing a fully-convolutional neural network built only from convolutional layers which is a very specialized type of network (typically used for segmentation tasks). 


Most networks will include other layer types which have trainable parameters too, i.e. batch normalization layers or densely connected layers. 
Yes, the training process operates on the same concept of error backpropagation.

If the model is performing a segmentation task (or any task which requires a 1-to-1 mapping of individually classified pixels), then yes its typically the same size.
The raw output of a fully-CNN is typically a probability map where each pixel is the probability between 0 and 1 of it being a member of the positive class. This is normally thresholded at 0.5 to obtain a binary mask which delimits the area of interest.
In essence, yes. But it may be more accurate to say that the goal is to minimise the loss function (reduce the error) which is done by learning optimal filter weights.
Overfitting is not about pixel location. In fact, CNNs are translationally invariant which means that the location of the feature has no impact on the output. There are some types of CNNs which take feature location into account but thats still not standard. Overfitting generally happens when a filter is an exact fit to a pixel patch, for example, the filter is no longer just looking for vertical lines but the exact pixel values of a section of vertical line in a specific image.
No, each convolution of a 3x3 filter takes 3x3 pixels and it is convolved over the whole image which means that the filter output will likely be around the same size of the image depending on stride length, padding and dilation. Have a look at one of my other answers for more details.